A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers (2018)

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C-1 APPENDIX C. DERIVATION AND VALIDATION OF QUASI-ELASTIC SIMULATION OF VISCOELASTICITY The objective of this appendix is to propose the concept of “quasi-elastic simulation of viscoelasticity” to enable accurate viscoelastic-to-elastic conversion using readily available material properties for asphalt pavement design. The following topics will be addressed in turn: 1. Definition of quasi-elastic simulation; 2. Physical significance of quasi-elastic simulation; and 3. Experimental validation of quasi-elastic simulation. Definition of Quasi-Elastic Simulation The concept of quasi-elastic simulation of viscoelasticity stems from the elastic- viscoelastic correspondence principle (1), which states that the solutions of a viscoelastic problem can be inferred from a reference elastic problem. Expressed using the constitutive equation, the viscoelastic stress and strain have the following relationship:       0 t ve d t E t d d         (C-1) where  ve t is the viscoelastic stress corresponding to the strain history  t ; t is the loading time;  is any arbitrary time between 0 and t ; and  E t is the relaxation modulus of the viscoelastic material. In the corresponding elastic reference, the constitutive equation becomes the relationship between the stress and the pseudo strain:  ve R Rt E  (C-2) where R is the pseudo strain; and RE is the reference modulus. A good option is to assign the elastic modulus, or Young’s modulus to the reference modulus (2). The resulting pseudo strain is thus expressed as: Undamaged state Damaged state e ve R e d            (C-3) where ve is the viscoelastic strain; e is the elastic strain; and d is the damage strain, which consists of the plastic strain, viscoplastic strain, and viscofracture strain. By definition, the elastic modulus is the ratio of the stress and strain that is instantaneous to this stress (i.e., elastic strain), namely it is time or frequency independent. As a result, when the elastic modulus is used as the reference modulus, the viscoelastic stress and resulting pseudo strain have a true elastic relationship that is independent of the frequency of the loading. Such a simulation can be regarded as the elastic simulation of viscoelasticity. In addition to the elastic modulus, for repeated loading another option is to use the dynamic modulus at the endurance limit as the reference modulus (3-4). The endurance limit is the threshold between the undamaged and damaged states under repeated loading. Determining the dynamic modulus of asphalt mixture at the endurance limit requires a series of tests at different loading levels and statistical analysis of measured material properties. Note that the dynamic modulus at the endurance limit remains statistically the same as the load repetition increases. When this dynamic modulus is adopted as the reference modulus, it yields the same pseudo strain as that in Equation C-3, but the difference is that the dynamic modulus is

C-2 frequency dependent. Therefore, the viscoelastic stress is elastically related to the resulting pseudo strain at a specific frequency but not at others. For a repeated load with different frequencies, different dynamic moduli must be used to compute the corresponding pseudo strain for asphalt mixtures. Compared to the elastic modulus, the dynamic modulus is more commonly used in asphalt pavement design to account for the frequency and temperature dependence of asphalt surface materials. Standard tests are available to measure the dynamic modulus under a haversine load. In the latest mechanistic-empirical pavement design methods, the dynamic modulus is utilized in the multi-layer elastic analysis to determine the primary responses in asphalt pavements. However, a recent study (5) brings up a question about such a usage: the standard method to calculate the dynamic modulus actually characterizes the material response to a sinusoidal load rather than a haversine load. A haversine load can be decomposed into a sinusoidal loading portion and a constant loading portion. To correctly represent the material response to a haversine load in the multi-layer elastic analysis, a combined compliance by the dynamic compliance and the creep compliance is proposed to characterize the relationship between the haversine stress amplitude and the resulting strain amplitude: * 1 0 0 2 2 p p f t t D D t                     (C-4) where 0 is the strain amplitude; 0 is the stress amplitude of a haversine load; *D is the dynamic compliance; f is the frequency of a load pulse; pt is the pulse time of a load; and  D t is the creep compliance. Based on Equation C-4, the appropriate compliance, or modulus, for the elastic analysis of asphalt surface materials should be the average of the dynamic response function at a frequency equal to 1 pt and the time-dependent response function at a time equal to 2 pt , called representative elastic compliance or representative elastic modulus. The representative elastic modulus is originally intended to be used in the multi-layer elastic analysis for pavement design. It performs a similar function as the elastic modulus but depends on the loading frequency. If the representative elastic modulus can be used as the reference modulus to compute the pseudo strain, it has practical implications for making use of the prevalent material property in asphalt pavement design, and avoid extra testing efforts devoted to determine the endurance limit. The following section explores this application to see whether the resulting pseudo strain has the same physical significance as that in Equation C-3. Using the representative elastic modulus to calculate the pseudo strain is called quasi-elastic simulation of viscoelasticity. Physical Significance of Quasi-Elastic Simulation The physical significance of the quasi-elastic simulation of viscoelasticity is discussed on the grounds of the theoretical derivation of the representative elastic modulus. The derivation of

C-3 the representative elastic modulus starts with a haversine-shaped strain pulse as shown in Figure C-1. The mathematical form of the strain pulse is:   0 sin 1 2 2 h h t t              (C-5) where  h t is the haversine strain pulse; 0h is the strain amplitude of the haversine pulse;  is the angular frequency; and  is the phase angle. Equation C-5 can be decomposed into two portions as:      1 2h h ht t t    (C-6) in which:   01 sin 2 2 h h t t           (C-7)   02 2 h h t   (C-8) Accordingly, the stress corresponding to  h t at an equilibrium state after the initial transient period is the sum of the stress causing  1h t and that causing  2h t , which is expressed as:      1 2h h ht t t    (C-9) in which:   *01 sin2 2 h h t E t         (C-10)    02 2 h h t E t   (C-11) where  h t is the stress corresponding to the haversine strain pulse; 1h is the stress causing  1h t ; 2h is the stress causing  2h t ; *E is the dynamic modulus; and  E t is the relaxation modulus. Then the magnitude of  h t is also the sum of two parts: *0 0 10 2 2 2 p ph h h f t t E E t           (C-12) The first part of Equation C-12 is based on the Pavement ME Design definition: the proper dynamic modulus or compliance to connect the stress and strain amplitudes is the one at a frequency equal to the inverse of the pulse time pt (6). The second part is based on the assumption stated in Underwood and Kim (5): the stress amplitude occurs when the strain pulse reaches a maximum at 2 pt . Equation C-12 further reduces to: *0 1 0 1 2 2 p ph f th t E E t               (C-13)

C-4 Therefore, the representative elastic modulus, denoted reE in this report, as the ratio of the stress amplitude to the strain amplitude, is calculated by: * 1 1 2 2 p p re f t t E E E t              (C-14) A further consideration is made herein for the assumption to obtain the second part of Equation C-12 mentioned above. It implies that when deriving the representative elastic modulus the stress and strain pulses are in phase, namely an elastic relationship exists between the stress and strain. This implication, though not pointed out in Underwood and Kim (5), is identical to the concept of the elastic-viscoelastic correspondence principle. As a result, the representative elastic modulus should be a promising candidate for the reference modulus. Next an experimental validation is conducted to further prove this inference. Figure C-1. Decomposition of Typical Haversine-Shaped Strain Pulse Experimental Validation of Quasi-Elastic Simulation The experimental validation is performed by comparing the calculated representative elastic modulus to the reference modulus measured from the test that resembles repeated traffic loading. It relies on previous research results on determining endurance limits of asphalt mixtures (3-4). The endurance limit is also called the critical nonlinear viscoelastic point, which serves as the reference state to quantify the damage. The dynamic modulus measured at this critical point, called the critical nonlinear viscoelastic property, successfully removes the viscoelastic effects when it is used to calculate the pseudo strain. The experimental validation makes use of previous laboratory testing results, which contain two types of laboratory testing:  Controlled-strain repeated direct tension test (RDT) (3, 7); and  Tensile creep and recovery test (8). The controlled-strain RDT tests are used to determine the critical nonlinear viscoelastic properties, denoted as *NLVEE , of tested asphalt specimens. They are conducted with 200 load cycles at a frequency of 1 Hz using the Material Test System (MTS) at 20°C. The value of

C-5 * NLVEE is calculated for representative load cycles (1st-10th, 50th-59th, 100-109th, 150th-159th, and 190th-199th) and the average of these cycles is used as the final result. The test setup, procedures and analysis methods were elaborated in the references above, so they are not repeated herein. The tensile creep and recovery tests are used to calculate the representative elastic moduli of the replicate asphalt specimens. They are performed using the MTS with a loading time of 60s and recovery time of 120s at three temperatures: 10°C, 20°C, and 30°C. The test setup was detailed in the reference above, so is not repeated here. Only the analysis method to obtain the representative elastic modulus is given as follows: 1) Calculate the relaxation modulus of each tested specimen at 10°C, 20°C, and 30°C, respectively, by the Laplace transform of the creep compliance measured from the test, the same method as used in Zhang et al. (2); 2) Construct the relaxation modulus master curve at the reference temperature of 20°C using the sigmoidal model suggested in the Pavement ME Design (6):   3 4 2 1 loglog 1 rr c c t cE t c e     (C-15) where rt is the reduced time of loading at the reference temperature; and 1c , 2c , 3c , and 4c are fitting parameters. 3) Convert the relaxation modulus master curve to the dynamic modulus by the following technique: a. Assume a Prony series form of the relaxation modulus as below and determine the Prony series’ coefficients by fitting the Equation C-16 to the master curve of the relaxation modulus constructed in step 2:   1 j tM j j E t E E e       (C-16) where E is the long term relaxation modulus; jE are the relaxation modulus coefficients; and j are the relaxation times. b. Compute the dynamic modulus by:   2 22 2 * 2 2 2 2 1 11 1 M M j j j j j jj j E E E E                              (C-17) With respect to the accuracy of the procedures above, efforts have been devoted to experimentally confirm that the dynamic modulus master curve converted from the creep test matches well with that measured from the dynamic modulus test. The materials used in the experimental validation are laboratory–mixed-laboratory- compacted hot asphalt mixtures, including two types of asphalt binder and one type of aggregate. The asphalt binders are designated “NuStar” with PG 67-22 from New Jersey and “Valero” with PG 64-16 from California. The aggregate is the Hanson limestone from New Braunfels, Texas. In order to produce asphalt mixtures with different material properties and behaviors, two air void contents (4-5% and 7-8%) and three aging periods (0, 3 months, and 6 months at 60°C) are selected during the mixture design. Thus, there are 12 types of asphalt mixtures. All of the twelve types of asphalt mixtures are subjected to the two kinds of tests mentioned above, and the results are given in Table C-1. Under “Mixture Type” of Table C-1,

C-6 the number “4%” or “7%” represents the air void content; the number of “0”, “3”, or “6” represents the aging period. In the controlled-strain RDT test, the dynamic modulus at the endurance limit, i.e. *NLVEE , is measured for each mixture type. In the tensile creep and recovery test, the value of 2 ptE t     (where pt = 1 s) is obtained from the relaxation modulus master curve. Figure C-2 shows two example master curves of the tested asphalt specimens. The value of * 1 p f t E  is calculated by Equation C-17 (where 2 f  and f  1 Hz). With known 2 ptE t     and * 1 p f t E  , reE is determined by Equation C-14. Plot *NLVEE versus reE in Figure C-3 and fit the data with a linear function. It proves that the critical nonlinear viscoelastic property is statistically equal to the representative elastic modulus. Therefore, when assigning the representative elastic modulus to the reference modulus, it yields the same result as using the critical nonlinear viscoelastic property to compute the pseudo strain. In other words, the quasi- elastic simulation removes the viscoelastic strain and produces the same pseudo strain as shown in Equation C-3. Figure C-2. Examples of Relaxation Modulus Master Curves at Reference Temperature 20˚C

C-7 Table C-1. Laboratory Test Parameters and Results for Validation of Quasi-Elastic Simulation Controlled-Strain RDT Test Test Parameters 1 f  Hz; 1pt  s; 6 .28  rad/s; temperature = 20°C Test Results Mixture Type *NLVEE (MPa) Mixture Type * NLVEE (MPa) NuStar, 4%, 0 5078 Valero, 4%, 0 7774 NuStar, 7%, 0 4519 Valero, 7%, 0 5320 NuStar, 4%, 3 7579 Valero, 4%, 3 10683 NuStar, 7%, 3 5055 Valero, 7%, 3 7823 NuStar, 4%, 6 10017 Valero, 4%, 6 12483 NuStar, 7%, 6 7432 Valero, 7%, 6 9044 Tensile Creep and Recovery Test Test Parameters Loading time = 60s; recovery time = 120s; temperature = 10, 20, 30°C Test Results Mixture Type 2 ptE t     (MPa) * 1 p f t E  (MPa) reE (MPa) NuStar, 4%, 0 4199 7030 5614 NuStar, 7%, 0 2963 5120 4041 NuStar, 4%, 3 5622 9740 7681 NuStar, 7%, 3 3554 5624 4589 NuStar, 4%, 6 7725 11048 9387 NuStar, 7%, 6 5269 8214 6742 Valero, 4%, 0 6501 10853 8677 Valero, 7%, 0 3691 6326 5008 Valero, 4%, 3 8425 12013 10219 Valero, 7%, 3 5801 9300 7550 Valero, 4%, 6 9690 13831 11761 Valero, 7%, 6 7673 11287 9480

C-8 Figure C-3. Comparison of Critical Nonlinear Viscoelastic Properties and Representative Elastic Moduli References: 1. Schapery, R. A (1984) “Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media.” International Journal of Fracture, 25, pp. 195-223. 2. Zhang, Y., Luo, R., Lytton, R. L. (2012) “Characterizing permanent deformation and fracture of asphalt mixtures by using compressive dynamic modulus tests.” Journal of Materials in Civil Engineering, 24(7), pp. 898-906. 3. Luo, X., Luo, R., Lytton, R. L. (2013a) “Characterization of fatigue damage in asphalt mixtures using pseudo strain energy.” Journal of Materials in Civil Engineering, 25(2), pp. 208-218. 4. Luo, X., Luo, R., Lytton, R. L. (2014) “Energy-based crack initiation criterion for viscoelastoplastic materials with distributed cracks.” Journal of Engineering Mechanics, 141(2):04014114. 5. Underwood, B. S., Kim, Y. R. (2009) “Determination of the appropriate representative elastic modulus for asphalt concrete.” International Journal of Pavement Engineering, 10(2): 77-86. 6. ARA, Inc. (2004). “Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures.” Final Report, NCHRP Project 1-37A. Transportation Research Board, National Research Council, Washington, DC. 7. Luo, X., Luo, R., Lytton, R. L. (2013b) “Characterization of asphalt mixtures using controlled-strain repeated direct tension test.” Journal of Materials in Civil Engineering, 25(2), pp. 194-207. 8. Luo, X., Luo, R., Lytton, R. L. (2013c) “Characterization of recovery properties of asphalt mixtures.” Construction and Building Materials, 48, pp. 610-621.